Jon Pretty

There are 100 prisoners, numbered 1-100. The prisoners' numbers are written onto 100 cards, put randomly into 100 envelopes numbered 1-100. They can't communicate, but are invited, one by one, to open 50 envelopes. If any prisoner fails to find his number, they all get killed.

At best, there's a 50% chance any one prisoner can find his number. It would seem like there's a probability of (½)¹⁰⁰ or 0.00000000000000000000000000008% that every prisoner could find his own number, if each were to check 50 envelopes at random.

They could certainly do even worse than this. If they all agreed in advance to open envelopes 1-50, then fifty of them would be guaranteed never to find their number and they'd all die. But could they come up with a strategy in advance which would improve their survival chances?

What makes this my favourite combinatorics problem is that there *is* a strategy which increases their likelihood of survival from 0.00000000000000000000000000008% to about 30%. So, what is the strategy, and why does it work?

Each prisoner can open 50, so the first has a 50% chance of killing everyone. Can the first leave the envelopes sorted in some way? Can the prisoners decide the order in which they go?